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Questions tagged [superalgebra]

For questions about superalgebra, which is a kind of graded algebra.

In mathematics and theoretical physics, a superalgebra is a $\Bbb Z_2$-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.

The prefix super- comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes.

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Grassmann numbers as eigenvalues of nilpotent operators?

The following question is motivated by the construction of the fermionic path/field integral, as done for example in Altland & Simons "Condensed Matter Field Theory". Consider the vector space $\mathbb C^2$ with two standard basis vectors named…
Greg Graviton
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What is the $\frac{1}{2}$ representation of $U(1)$?

This may be a silly question, and so I apologize in advance. But it stems from a reading of section 2 (page 5) of the physics paper, "Counting chiral primaries in N=1 d=4 superconformal field theories". My question is: What does the representation…
leastaction
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Orthosymplectic Lie Superalgebra

I am trying to work out a presentation for the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2n)$. I am following Musson's book "Lie Superalgebras and Enveloping Algebras". From what I understand, we can take as the underlying…
freeRmodule
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A good reference for learning about super-differentiation & super-integration?

I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear set of simple examples to build up some basic…
Mozibur Ullah
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Is Whitehead lemma true for super Lie algebras?

Classical Whitehead lemma states that if $\mathfrak g$ is a finite-dimensional complex Lie algebra and $M$ is a finite-dimensional $\mathfrak g$-module, then first cohomology group $H^1(\mathfrak g, M)$ (defined for example as cohomology of the…
Blazej
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What is contragredient about contragrediant Lie superalgebras

Among the Lie superalgebras there is the class of contragredient Lie superalgebras. Roughly speaking these are those Lie superalgebras that can be defined with a matrix $a_{ij}$ and commutation relations of the generators $H_i,E_i$ and $F_i$ which…
Buzz Lee
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A Grassmann-Variable Identity from Wikipedia

I found this identity on Wikipedia: $$\int\exp\left[\theta^T A\eta+\theta^T J+K^T\eta\right]d\theta d\eta =\det A\exp\left[-K^TA^{-1}J\right]$$ where the integration variables are Grassmann variable. Unfortunately the Wikipedia page gives little…
Jessica
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Geometric interpretation of Supersymmetry

Is there a geometric interpretation of supersymmetry? I.e., if one has a manifold $\mathcal {M} $, and there are $\mathcal {N} $ SUSY generators, then is there a geometric interpretation of the SUSY generators?
user122283
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A question about super vector space

Let $V=V_0\oplus V_1$ be a superspace, where its even subspace $V_0$ and odd subspace $V_1$ both are $k$-dimensional. Let $(e_1,\cdots,e_k|f_1,\cdots,f_k)$ be a basis of $V$. Consider linear transformations $\varphi,\psi$ on $V$, such that…
qinr
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Super tensor product as paracomplex tensor product

Consider a ground field $k$ of characteristic zero, and the category of super vector spaces $\mathrm{SVect}_k$ with the grading $V = V_0 \oplus V_1$ and homomorphisms $f: V \to W$ satisfying the conditions $f(V_i) \subseteq W_i$. Equivalently,…
Aleksei Averchenko
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Definition of $\mathbb{Z}_2$-graded module homomorphism

A $\mathbb{Z}_2$-graded module homomorphism $f$ of a $\mathbb{Z}_2$-graded algebra $A$ is defined to satisfy $f(am)=(-1)^{\text{deg}(f)\text{deg}(a)}af(m)$, for example, at the bottom of the second page of the paper [J]. When $f$ and $a$ have…
edittide
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Wedderburn theorem version for superalgebras

I am looking for an example of usage of wedderburn theorem version for superalgebras (which is a $\mathbb{Z}_2-$graded algebra). The theorem states that if $ A$ is finite dimensional $\mathbb{Z}_2-$graded simple associative algebra (it can be…
matan
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$\mathbb{Z}_2-$grading by Hodge star operator.

Consider the algebra of exterior forms $\Lambda T^*M$ on an even dimensional $n-$manifold $M$. We can form an operator $\sigma=\bigoplus_{k=1}^ni^{k(n-k)}*_k:\Lambda T^*M\rightarrow\Lambda T^*M$ (where $*_k$ is the Hodge star restricted to…
Guest123412341234
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The super group $GL(1|1)$

It is difficult to find information on super groups and I have built my knowledge from various sources. I have the following questions. $GL(1|1)$ is defined as the group of invertible linear transformations on $\mathbb{C}^{1|1}$. I see conflicting…
Okazaki
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Supercommutative algebras except commutative algebras and exterior algebras

When I first saw a definition of a supercommutative algebra, first example that came to my mind was an exterior algebra on some vector space. Of course, purely even supercommutative algebra is just a commutative algebra, so we count commutative…
ante.ceperic
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